investigation of different aspects of laminar horseshoe vortex system using piv

Journal of Mechanical Science and Technology 28 (2) (2014) 527~537

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-013-1120-9

Investigation of different aspects of laminar horseshoe vortex system using PIV†

Muhammad Yamin Younis1,*, Hua Zhang1, Bo Hu1, Zaka Muhammad1 and Saqib Mehmood2

1National Key Laboratory of Fluid Mechanics, School of Aeronautical Science and Engineering, Beihang University (BUAA), Beijing, 100086, China 2School of Astronautics, Beihang University (BUAA), Beijing, 100086, China

(Manuscript Received October 11, 2012; Revised July 29, 2013; Accepted August 21, 2013)

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Abstract Juncture flow is a classical fluid mechanics problem having wide applications in both aero and hydro dynamics. The flow separates

upstream of the obstacle due to the adverse pressure gradient generated by it, with the formation of the vortical structure called “horse-shoe vortex.” The current study is carried out for an elliptical leading edge obstacle placed on a flat plate to investigate the horseshoe vortex for a range of Reynolds number (ReW) based on maximum width (W) for which the incoming boundary layer is laminar. Four different types of horseshoe vortex systems were found: the steady, amalgamation, transition and breakaway. The transition vortex sys-tem is one after which the vortex system changes from amalgamation to breakaway. In this phase the vortex system alternatively under-goes both amalgamation and breakaway vortex cycles. The effect of variation in the chord wise shape of the obstacle is investigated. The quantitative measurements of PIV show that the vortex system does not undergo any significant change for different chord lengths of the model with the fixed aspect ratio and maximum width. The most upstream saddle point is also studied for steady horseshoe vortex region and found that it is the “saddle of attachment” where flow attaches to the plate surface instead of separating from it.

Keywords: Juncture flows; Transition vortex system; Chord length; Saddle of attachment; PIV ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

A commonly experienced phenomenon in various aerody-namic and hydrodynamic applications is that of junction flows. When the flow attached to a flat surface encounters an obsta-cle, the boundary layer separates from it. The cause of this flow separation is an adverse pressure gradient in the upstream region of the juncture. This incoming separated flow then rolls up in the space to form a vortical structure known as a “horse-shoe vortex.”

The laminar horseshoe vortex has been studied for a long time. The earliest study was carried out by Schwind [1] who found the horseshoe vortex pattern upstream of the obstacle placed on a flat surface. The vortex pattern changed with the increase in velocity, exhibiting a steady vortex system with further increase in velocity changes to oscillatory behavior and then to a periodically shed vortex system. Norman [2] from flow visualization proposed the steady horseshoe vortex pat-tern. Baker [3] found that with the change in Reynolds num-ber the steady horse shoe vortex changed to unsteady ones. He also presented the topological sketches for laminar steady horseshoe vortex system. Baker [4] explained the reason for the vortex oscillation and proposed that vortex core oscilla-

tions are due to oscillations of the whole vortex system and vortex core instabilities. Visbal [5] provided computational results for a circular cylinder at Reynolds numbers from 500 to 5400. He presented a new topological picture for separation point upstream of cylinder. Khan [6] visualized the vortex structure for a wing model with elliptic (3:2) leading edge and NACA 0020 trailing edge for Reynolds number ReT ≤ 6000. He categorized the horseshoe vortices into steady, oscillating, and shedding and splitting mode. Seal et al. [7] suggest that instability in the impinging shear layer downstream of the attachment saddle point probably determines the breakaway or shedding frequency. Zhang et al. [8] found that the increase in the angle of attack causes the horseshoe vortex to change from single to multiple vortex system, and exhibits unsteady char-acteristics instead of steady ones. On the other hand the in-crease in swept angle reduces the strength of the horseshoe vortex system. Wei et al. [9] classified the horse shoe vortex system into steady, periodically oscillating, unsteadily oscillat-ing and turbulent vortex for a square cylinder juncture using flow visualization.

Simpson [10] reviewed the work undertaken to study the structure of horseshoe vortex in both the laminar as well as turbulent flow regime along with the effect of different pa-rameters on vortex structure and about its control. Lin et al. [11] classified the horse shoe vortices into four different cate-gories steady, periodic oscillation, periodic breakaway and

*Corresponding author. Tel.: +86 1082331164, Fax.: +86 1082313570 E-mail address: [email protected]

† Recommended by Associate Editor Simon Song © KSME & Springer 2014

528 M. Y. Younis et al. / Journal of Mechanical Science and Technology 28 (2) (2014) 527~537

turbulent like vortex system and specified the range of differ-ent vortex regimes for a vertical-plate base plate juncture. Lin et al. [12] also studied the square cylinder flat plate juncture and classified the horseshoe vortex system. Khan [13] pro-vided a topological model of flow regime for the unsteady flows, i.e., oscillating, and shedding and breakaway vortex system. Rodríguez et al. [14] studied the horseshoe vortex confined between the flat plates and analyzed the effect of Reynolds number on horseshoe vortex; they also studied the effect of aspect ratio on horseshoe vortex structure strength and found that with the increase in aspect ratio the strength of horseshoe vortex increases; as it reaches a certain value the structure then becomes independent of aspect ratio. The effect of aspect ratio on horseshoe vortex was also studied by Rifki [15] who found that the aspect ratio plays a very important role in mixing and heat transfer and also has a strong correla-tion with the horseshoe vortex structure. Wei et al. [16] also studied the variation in the vortex structure with the alteration in the leading edge shape of the obstacle. They found that the horseshoe vortex is of smallest size and weakest in strength for sharp leading edge (triangular) cylinders and most strong and bigger for flat (square) leading edge cylinders. Zhang et al. [17] using quantitative flow visualization technique of PIV studied the horseshoe vortex structure and categorized the unsteady vortex system in three modes: amalgamating, shed-ding off amalgamating and shedding off dissipating. They also explained the shedding off process and explained that the sec-ondary vortex tongue isolates the primary vortex from imping-ing boundary layer and sheds it off from the upstream vortex region. Younis et al. [18] studied the laminar horseshoe vortex for a sharp leading edge (triangular) cylinder and found a range of steady as well as unsteady vortex region for it.

The boundary layer separation point in the juncture flows is a very important problem that is still under discussion. Ex-perimental, theoretical as well as numerical simulations were conducted to study the topological pattern of the horseshoe vortex and to understand the nature of the most upstream sad-dle point. Traditionally Maskell [19], Lighthill [20] and Wang [21, 22] considered the 3-D boundary layer separation as a phenomenon in which boundary layer separates from the solid surface and forms a spatial vortex sheet under the adverse pressure gradient. Baker [3] made cartoons of topology for two, four and six vortexes, steady laminar horseshoe vortex system, all of them exhibit traditional separation (when skin-friction-lines converge the limit streamline may rise up from the surface to form the spatial vortex). As per his conclusion the outmost vortex separates from the plate surface at half saddle point and this 3-D separation point is called “separation saddle point.” Visbal [23], contrary to traditional separation, by his numerical simulations revealed a different vortex topol-ogy in the juncture symmetric plane. The streamlines, instead of rising up from the surface attach to it, which he named the most upstream saddle point as “attachment saddle point.” Hung et al. [24], Chen et al. [25], Puhak, et al. [26], Rizaetta [27], and Chen [28] also got attachment saddle point from

their simulations. Coon and Tobak [29] also found attachment saddle point structure from their visualization results. Recently, Zhang et al. [30] also confirmed the existence of attachment saddle point structure. They also explained three attachment saddle point topologies and provided the topological pictures in the symmetry plane for steady laminar horseshoe vortex system and the experimental results confirming these three topologies. Unlike Baker [3] and Visbal [24] they explained these topological structures in symmetry plane and in the sur-face plane of the vortex separately.

The experimental results are still limited to understanding a number of things in the juncture flows, such as the effect of boundary layer parameters, effect of chord length on the vor-tex structure, and the nature of singular point etc.

The current study is carried out to study the horseshoe vor-tex for a 1:2 elliptical leading edge cylinder with the change in Reynolds number, analyze the effect of the chord length of the obstacle on the nature of the laminar horse shoe vortex, and investigate the singular point characteristics of steady laminar horseshoe vortex system in the symmetric plane.

2. Experimental setup

The experiments were conducted in a low speed water channel at National Key Laboratory of Fluid Mechanics, in the Fluid Mechanics Institute of Beihang University, with test section size of 400×400 mm2 and length of 4m. A flat plate with 5:1 elliptic leading edge and length of 1500 mm was installed in the water channel. Three elliptical models all hav-ing elliptical (a:b = 1:2) leading edge as shown in Fig. 1 with major axis a = 50 mm and minor axis b = 25 mm having height H = 250 mm (A.R = 2.5 where W = 2a) were mounted on the flat plate to generate the juncture flow. The end of cyl-inder was submerged in the water at least 5cm below the free surface of the water to prevent the influence of the end. The free stream velocity ranged from 1-9 cm/s, so that the experi-mental Reynolds number defined by cylinder maximum width (W) ranged from ReW = 1000~7000; within this range of Rey-nolds number the juncture flow exhibited laminar three-dimensional horseshoe vortex characteristics. The Reynolds number based on displacement thickness of the boundary layer (Reδ*) experimented ranged between 470 ~Reδ*~930.

To study the effect of the chord length of the models (obsta-cle) on horseshoe vortex structure two more models along with the one mentioned above were investigated, all having similar leading edge (1:2 elliptical) shape and similar maxi-mum width “W”. The A.R is also same (2.5) but with varying chord lengths as shown in Fig. 1. From some previous ex-perimental (unpublished) results in the lab it was observed that the cross sectional changes downstream of the maximum width of the modal do not have any influence on the symmet-ric plane vortex structure upstream of the juncture, so here in this study model 1, which is without sharp trailing edge, does not make any difference from that of the other two models with trailing edge. The experiments were carried out using a

M. Y. Younis et al. / Journal of Mechanical Science and Technology 28 (2) (2014) 527~537 529

2-D PIV system which consists of dual-cavity pulse Nd:Yag laser with Δt = 200 ns~600 ns and power of 200 mJ/Pulse, laser sheet arm, laser pulse synchronizer and CCD camera with spatial resolution of 2448 × 2050 pixels and with maxi-mum data sampling frequency 30 frames/s. A MACRO 105 mm F2.8 EX DG lens was used to get a better detailed view of the flow structure. Aluminum di Oxide Al2O3 particles with diameter of 400 nm and the specific gravity of about 3.4- 4.0 were seeded to track the flow structure in front of the juncture.

The post-processing of the visualization data used in this study was done using the world technology development co. Ltd. Beijing Cubic Micro Vec V3.3.1 Software integrates the PIV, PTV, concentration and particle size analysis and field analysis module. The PIV system is shown in Fig. 2. The ex-perimental setup and PIV measurement zone are shown in Fig. 3. Overall, a region of 110×30 mm2 was considered for the

analysis, which then was divided into small regions for differ-ent flow analysis.

The model was placed 500 mm downstream of the leading edge of the plate in all the experiments as shown in Fig. 3. The flow Reynolds number was well below the transition Rey-nolds number; also boundary layer thickness δ on flat plate was measured for all the experimented flow velocities prior to placing the model on it for the same location where later on the leading edge of the model was positioned. A comparison of the boundary layer thickness obtained from the experiments with that of the Blasius for few flow velocities is presented in Fig. 4, which shows very good agreement between the two.

3. Results and discussion

3.1 Vortex structure in front of 1:2 elliptical leading edge cylinder

The structure of the horseshoe vortex was studied for the range of Reynolds number (1000 ≤ ReW ≤ 7000); the cylinder with maximum thickness W = 100 mm (Model 1) was used for this study. The results obtained show similar phenomenon in steady and unsteady laminar behavior of the horseshoe vortex as obtained by previous studies. The vortex structure is divided into four regimes: the steady, amalgamating, transi-tional (both amalgamating and breakaway cycle exists alterna-

Fig. 1. Experimental models configuration.

Fig. 2. PIV system.

Fig. 3. Experimental setup and PIV measurement zone.

Fig. 4. Comparison for experimental and blasius boundary layer profiles.


tively) and breakaway horseshoe vortex system. The flow structure was found to be steady for the Reynolds

number ReW < 2500. For all these Reynolds numbers the horseshoe vortex was found to be fixed with no observable displacement or oscillation of the vortex structure with time, contrary to the fact of an increase in number of primary vor-tices as the Reynolds number was increased. The particle trac-es for the single, double and triple primary vortex structure are shown in Fig. 5. In Fig. 5(a) there is only one primary vortex, with clockwise rotation; in Figs. 5(b) and (c) the two and three primary vortices exist, respectively, all with clockwise rotation.

For closer analysis, the experimental section was divided in-to two small sections with (55 mm×30 mm) each, so as the changes or the unsteadiness in the vortex structure can be more carefully observed. Fig. 6 shows the most downstream (closest to the leading edge of cylinder) primary vortex and part of the primary vortex upstream of it. The PIV data was collected for 50 seconds and the analyzed using Matlab ® for the velocity response with time for all the cases till the insta-bility in the flow appeared. For this investigation, the velocity time response at three different points was observed. The three points were chosen around the most downstream (near the leading edge of obstacle), primary vortex for being the most sensitive to any slight change in the vortex system. The veloc-ity time response for three points marked in the streak line image as above, center and below the vortex core in Fig. 6 are shown above it, clearly portrays that for all three points, above, below and at the center of the vortex core, no fluctuation in velocity magnitude with time is observed. Similar velocity time response is observed for all case (ReW < 2500).

From the investigation, unsteadiness appears as the Rey-nolds number ReW > 2500; the flow structure becomes un-steady with small oscillation. The structure is of amalgamation type at low Reynolds number, which then changes to break-away vortex system after going through the transition phase. Baker [3] divided the unsteady laminar oscillating vortex sys-tem as low frequency oscillation mode and high frequency oscillations mode. Khan [6] named both modes as oscillatory mode and shedding and dissipating mode respectively, whe-reas Lin [11] named them as amalgamation and breakaway vortex system, respectively. The former (amalgamation) vor-tex system is described as the one in which the most down-stream primary vortex tries to escape the upstream vortex structure with downstream motion towards the obstacle, but as it moves towards the obstacle, under the adverse pressure gradient of obstacle and the shear of the flat surface, it dissi-pates and cannot continue its movement. The dissipation of the vortex makes it weaker and at last is sucked by the strong-er upstream vortex system, and becomes the new vortex which again will undergo the same cycle of dissipation and amalgamation. The latter (breakaway) vortex system is de-fined as the one in which the most downstream primary vortex splits from the upstream vortex system and moves towards the obstacle. During its movement towards the obstacle, unlike the amalgamating vortex system, it continues its motion and

reaches quite near to the obstacle. But it cannot cross the ob-stacle and stops near it and keeps on dissipating. Its size con-tinues to decrease with increase in its rotational velocity; meanwhile another primary vortex from upstream approaches the remaining part of dissipated vortex and engulfs with it and this new vortex again keeps on rotating in a similar way as explained earlier until a new upstream primary vortex ap-proaches and engulfs with it. The upstream vortex structure in all the unsteady laminar vortex systems remains the same, with continued shedding of primary vortices, starting from most upstream singular point. Depending on the vorticity of incoming boundary layer with changing Reynolds number, the behavior of the most downstream (near the cylinder leading edge) primary vortex, changes and forms different kinds of

Fig. 5. Steady laminar horseshoe vortex system.

Fig. 6. Velocity fluctuation at three different position for a steady horseshoe vortex system (ReW = 1970).


unsteady vortex systems. In the present study also, the two modes were observed, the low frequency mode (amalgama-tion) is observed for the Reynolds number ReW = 3333 and ReW = 3930, and high frequency mode (breakaway) is found for ReW = 5075, 5523, 6070, 6468, and 6966. The shedding frequency is found increasing with increasing Reynolds num-ber, a characteristic of all unsteady horseshoe vortex systems. Velocity variation with time and power spectrum for the two

modes at two specific Reynolds numbers is shown in Figs. 7 and 8.

An important phenomenon found in the present study is of transition vortex system. Two Reynolds numbers observed between the low and high frequency mode, ReW = 4280 and 4580, the vortex system were both amalgamating and break-away, changing from one to other alternately. One such cycle is shown in Fig. 9. In Fig. 9(a) the primary vortex nearest to the leading edge of the cylinder is represented as P1; when it moves towards the leading edge of the cylinder it undergoes the break-away process as shown in Figs. 9(a)-(e).

When the primary vortex completely breaks away a new primary vortex replaces this vortex: P2, Fig. 9(e). The primary vortex which already broke-off now continues its motion to-wards the cylinder and during this movement it dissipates (Figs. 9(f)-(i)), its size reduces greatly and eventually it dissi-pates completely (Fig. 9(j)) under the influence of shear be-tween vortex and flat plate.

At this stage of the shedding process the new primary vortex P2 now replaces the former P1 vortex; this new vortex P2 instead of followig the same track as followed by its predecessor, moves slightly downstream towards the cylinder (Figs. 9(k)-(m)) and then is sucked by the upstream vortex system (Figs. 9(n)-(p)), undergoing the amalgamation process instead of breakaway. On accomplishment of this amalgamation process, the new primary vortex (Fig. 9(q)) then starts moving downstream represented by P1* (Figs. 9(r)-(t)) and follows the same track as followed by the primary vortex P1 earlier and undergoes the breakaway process. The vortex system keeps on undergoing breakaway and amalgamation alternately.

The velocity variation at ReW = 4280 for a period of 300 seconds is measured at two different locations upstream of cylinder. The positions where the velocity variations are measured are at (39, 2) and (19,2) upstream of leading edge of cylinder. The velocity variation for position (39, 2) is labeled as Location 1 and for position (19, 2) is labeled as location 2 in Fig. 10. Location 1 is selected here in such a way that it is close to the vortex region from where the most downstream primary vortex starts its movement downstream towards the cylinder leading edge either to go for amalgamation or break-away. Location 2 is selected near to the leading edge of the cylinder. Both locations are 20 mm apart from each other.

According to the velocity variation plot with time, shown in Fig. 10, the results show that the velocity at location 1 (repre-sented with the red, solid line) undergoes for two cycles for the same time for which the velocity variation of one cycle is observed at location 2 (blue line with a square dot), and this cyclic behavior continues with time. Now from the plot (Fig. 10), for each of the amalgamation and breakaway cycles the velocity varies at location 1 and can clearly be observed (red solid line), but for location 2 velocity variation mainly due to the breakaway process is clearly observed, with a small or negligible response due to the amalgamation process. The power spectrum for this transitional behavior of vortex system

Fig. 7. Velocity fluctuation and power spectrum for amalgamating vortex system.

Fig. 8. Velocity fluctuation and power spectrum for breakaway horse-shoe vortex system.


is also observed and shown in Fig. 10. This shows that the shedding frequency at location 1 is twice at location 2. The apparent reason of this is that location 2 does not undergo any observable variation of velocity for the amalgamation vortex system. The input free stream velocity is constant throughout the course of the experiment, so the measurement of the power spectrum (Fig. 10) could give a direct measure of the system’s modal characteristics. Hence the dominant peaks in Fig. 10 could be taken as a representation of its respective modes. From Fig. 10, the most dominant peak (taken as mode 1) at location 1 occurs at 0.278 Hz, whilst for location 2 it occurs at 0.14 Hz. It is interesting that mode 1 at location 2 occurs at the same frequency as mode 2 for location 1. An important observation from this may be made: during the tran-sitional process the vortex shedding is observed to be of some fixed primary shedding frequency and is independent of the nature of the vortex shedding (amalgamation or breakaway)

Fig. 9. Transition vortex system.


and that the vortex shedding at different positions is equal to some mode of primary shedding frequency. Though the vortex system undergoes alternately for amalgamation and break-away the frequency at two different locations are having some relationship between each other.

3.2 Effect of chord length on horseshoe vortex

Several factors based on the obstacle shape or size may af-fect the horseshoe vortex structure, e.g., aspect ratio, the lead-ing edge shape etc. In the present study the effect of the chord length of the obstacle is studied for the laminar horseshoe vortex system. The study is carried out with three models sim-ilar in leading edge shape (1:2 elliptical) and with same aspect ratio the three cross sections are shown in Fig. 1, and labeled as Model 1, Model 2 and Model 3. Models are placed in such a way that the leading edge of them is always positioned at the same downstream position from the leading edge of the plate (X = 500 mm).

The PIV results shown in Figs. 11 and 12 are for three chord lengths, 125 mm, 225 mm and 325 mm, at two different Reynolds numbers ReW = 1970 and 2390, respectively.

Visualization outcomes at Reynolds number ReW = 1970 (Fig. 11) reveal that for all three chord lengths the flow struc-ture is steady horseshoe vortex system with three vortices (two clockwise rotating primary vortices and one counterclockwise rotating secondary vortex). Furthermore, for all the cases the position of the vortex cores is approximately at the same hori-zontal distance upstream of the cylinder leading edge. For the most downstream primary vortex this position for all three cases is compared with a vertical yellow dashed line Fig. 11;, all the vortex cores lie approximately at the same horizontal position.

At Reynolds number of ReW = 2390, the horseshoe vortex

system is observed to consist of four vortices; out of them three are clockwise rotating primary vortices, but one counter-clockwise rotating secondary vortex. The structure is similar for all three chord lengths as shown in Fig. 12. Again, the horizontal positions of the vortex cores upstream of the cylin-der leading edge for all vortices in each of three chord lengths are similar to each other. For the most downstream primary vortex the horizontal position is compared for all three cases (Fig. 12) with a vertical yellow dashed line, which confirms the previous statement.

Three more Reynolds numbers are selected for this com-parison--ReW = 4280, 5075, and 6070--for each of the three chord lengths. For all of the three cases the vortex system is found to be laminar unsteady horseshoe vortex system. In the

Fig. 10. Velocity variation and power spectrum at two positions for transition vortex system (ReW = 4280).

Fig. 11. Horseshoe vortex system at ReW = 1970 (comparison for three chord lengths).

Fig. 12. Horseshoe vortex system at ReW = 2390 (comparison for three chord lengths).


first case at ReW = 4280 the vortex system is of transitional, as explained in the above section; the vortex system alternately exhibits amalgamation and breakaway behavior. This situation also persists for all three cases irrespective of the chord length. The velocity variation for all the chord length is measured at location (39, 2) to compare the velocity variation and further to compare the Strouhal number. The velocity variation and corresponding power spectrum is shown in Fig. 13; the veloc-ity fluctuations in all three cases at the same location show similar behavior and approximately same magnitudes.

Again, the power spectrum (Fig. 13) also reconfirms the behavior of the vortex system for three different chord lengths. The dominant frequency peak is observed at (St. no = 0.2839 Hz), and for three chord lengths the frequency peaks overlap each other at the same value.

At even higher values of Reynolds number, ReW = 5075 and ReW = 6070, the horseshoe vortex system now changes from transitional to completely high frequency or breakaway mode. This situation continues in all three cases under observation (chord length 125 mm, 225 mm and 325 mm). In Fig. 14 the velocity variation and power spectrum data for three cases at ReW = 5075 is presented. The velocity variation curves show that for a period of as long as 180 seconds the data exhibits a constant time period for each cycle of the breakaway vortex system irrespective of the variation in the chord length. The velocity magnitudes are also nearly the same for all cases. PSD, in Fig. 14, also displays a sharp dominant peak at Strou-hal number of (St. no = 0.3734) for all three chord lengths. The results are found to be repeating in similar fashion as the Reynolds number is increased to ReW = 6070, but with a higher nondimensional shedding frequency (St. no = 0.436) for all cases (chord lengths) under observation.

3.3 Saddle of attachment and topology of laminar steady

horseshoe vortex system

Topological study of the separation point is a hot issue in juncture flows. A number of studies [24-31] show that the most upstream singular point in 3-D separation in juncture flows is saddle of attachment where flow attaches to the plate surface instead of the traditional saddle of separation [19-22], the case where boundary layer lifts off from the surface of plate to form a vortical structure.

Visbal’s [23] numerical simulations provided the topologi-cal picture for such attachment saddle point in the case of steady laminar horseshoe vortex flow. Further computational and experimental results [24-31] supported this idea and pro-vided the theoretical basis along with their studies. A recent study by Zhang [30] provided a detailed description of the 3-D singular point for laminar horseshoe vortex; he also explained from PIV experimentation, the existence of different topologi-cal cartoons for the same vortex system in the symmetry plane, naming them as attachment topology A, attachment topology B and attachment topology C. This experimental confirmation of singular point structure as attachment saddle point in 3-D

separation in juncture flows by Zhang [30] using PIV further strengthens the already available studies on the topic.

Now for the case of elliptical model, with chord length (C.L = 125 mm) the singular point is also observed in the present study. As mentioned before that for ReW = 1075 the vortex system is steady with a single primary vortex (N), a space saddle (S) a half attachment node (NA’), where the flow at-taches to the plate surface and a half attachment saddle (SA’) near the cylinder in the symmetry plane also as explained by Zhang [30]. The above described details are presented in Fig. 15 with both the particle streak lines and streamlines for the horseshoe vortex system. Due to the low resolution of the camera used, the small corner vortex is not observed clearly.

Fig. 13. Comparison of variation in velocity magnitude and vortex shedding frequency for unsteady horseshoe vortex system for three different chord lengths at ReW = 4280.

Fig. 14. Comparison of unsteady vortex (breakaway) behavior at ReW

= 5075 (velocity variation and power spectrum) for three chord lengths.


At ReW = 1700 the vortex system in symmetry plane now has two primary vortices (N) with two space saddles (S); the flow again is observed to be attaching to the surface at a half node of attachment (NA’) rather than lifting from the surface as in traditional 3-D separation, along with a half attachment saddle point(SA’) near the cylinder also observed in the Fig. 15. At ReW = 2400 the vortex system now consists of three primary and one secondary vortex (N) with three space saddles (S). The flow is observed to be diverting from the most upstream space saddle (S) and attaching to the surface with a half node of attachment (NA’) in symmetry plane, which again reconfirms the existance of attachment saddle point structure of the 3-D flow separation. A half saddle of attachment (SA’) near the cylinder leading edge is also observed. A secondary vortex (N) initiates from the surface at half saddle of separation (SS’) and again attaches to it at half saddle of attachment (SA’).

In all three cases due to the experimental limitations no

clear counter rotating secondary vortices (except one secondary vortex in case of ReW = 2400) and corner vortices are observed.

All three cases explained above exhibit the characteristics of attachment saddle point topology explained by Visbal [23] and attachment topology B explained by Zhang [30]. The difference between the descriptions of the two is that the for-mer explains the 3-D separation considering both the symme-try plane and surface plane simultaneously using computation, whereas the latter just considers the symmetry plane to explain the 3-D laminar steady separation with PIV experimentation. It is difficult to observe the skin-friction line pattern on the surface of the plate using quantitative methods such as PIV due to the very small flow rates. Zhang et al. [30] used compu-tational results of Wang [31] to correlate the symmetry plane topology of the attachment node to that of the surface plane to observe the combined flow topology in two planes. This is also important to understand how the two different symmetric plane flow patterns are different from each other in surface plane. Do they have similar surface plane flow structure or have some variance? Study similar to that of Wang [31] is carried out for the current case of elliptical leading edge model. The computational method and other details are used similar to that of Wang [31] and are not discussed here. The stream-line pattern in the symmetry plane as well as skin-friction lines pattern on surface plane are shown in Fig. 16. In the symmetry plane for the computed vortex system (Fig. 16) the topology is identical to that of single primary vortex horseshoe vortex system ReW = 1075 (Fig. 15). But the comparison of com-puted (Fig. 16) and experimental result (Fig. 15) shows that the skin-friction lines flow pattern in surface plane is a con-vergent asymptote (normally a flow separation characteristic), and not the divergent one (a feature usually associated with the flow attachment).

On the basis of the results from experiments and computa-tions the 3-dimensional topological cartoon for the single pri-mary vortex horseshoe vortex system, with attachment saddle point topology for both symmetry plane and surface plane can be inferred. To analyze the 3-Dimensional attachment saddle point topology and to compare it with traditional separation saddle point topology, the two topologies are described in Fig. 17. Although the two topological pictures in the symmetry plane exhibit different structures, they correspond to identical skin friction lines on the wall: the skin-friction-lines on the plate surface will converge to the asymptotic line which origi-nates from a saddle point on the plate surface (Fig. 17), which shows the three-dimensional picture of two flow structures. Fig. 17(a) corresponds to the traditional separation saddle point structure, whereas Fig. 17(b) corresponds to attachment saddle point structure. It is difficult to assess the spatial flow topology only on the basis of the skin-friction-lines obtained from the surface oil flow visualization--that whether it is con-ventional topology or a new topology. The conclusion thus deduced is also similar to that of Visbal [24] that the Skin-friction-line's convergent asymptote is only a necessary condi-

Fig. 15. Attachment half node point structure in symmetry plane for steady laminar horseshoe vortex system.


tion but not a sufficient condition of 3-D steady separation; and second, as spatial flow attaches to plate surface the skin-friction-line may not present a divergent asymptote line but may be a convergent asymptote line.

Thus, it can be concluded that the outermost primary vortex in case of laminar horseshoe vortex system may originate from a space saddle (attachment topology) instead of a surface saddle (traditional separation saddle point), and also the skin-friction convergent asymptote may also represent an attach-ment line instead of a separation line. Zhang [32] called this convergent asymptote as the secondary type of attachment line. This experimental study further reconfirms the results of Vis-bal [23] and many others [24-31] about the existence of at-tachment saddle point structure in the horseshoe vortex system.

4. Conclusions

The quantitative measurements of PIV show that for ellipti-cal leading edge cylinder with ratio (1:2), the horseshoe vortex system undergoes steady vortex system till ReW = 2500, for higher Reynolds number and then converts to unsteady lami-nar vortex system. The vortex system oscillates alternatively in both amalgamation and break away fashion for a short range of Reynolds number as it changes from the former re-gime to later one, exhibiting two different shedding frequen-cies at two different points, but the two different frequencies are modes of some specific fundamental frequency.

The results also indicate that the chord length (C.L.) or any change of geometry of the cylinder downstream of the maxi-mum width (maximum width of cylinder) does not have any effect on the characteristics of horseshoe vortex in symmetric plane. The maximum width location from leading edge of obstacle, thus is the critical length on which the horseshoe vortex depends, downstream of which neither the chordwise extension of model nor its geometry plays any role on sym-metric plane structure of horseshoe vortex system.

From the results, the most upstream singular point in case of steady laminar horseshoe vortex system is again found to be saddle of attachment rather than the saddle of separation. The flow attaches to the surface instead of leaving the surface, confirming that the vortex initiates from a space saddle instead of from a surface saddle point. Reconfirming the existence of attachment saddle point topology.

Acknowledgment

This research is supported by National Science Foundation of China (NSFC), Grant Number: 11372027.

Nomenclature------------------------------------------------------------------------

a : Major axis of leading edge of model A.R : Aspect ratio b : Minor axis of leading edge of model N : Node or focus NA’ : Node of attachment S : Saddle SA’ : Saddle of attachment SS’ : Saddle of separation PIV : Particle image velocimetry ReW : Reynolds number based on maximum width of model Reδ* : Boundary layer displacement thickness Reynolds

number St. no : Strouhal number W : Maximum width of model

References

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Fig. 16. Simulation result of attachment topologies of circular cylin-der/flat plate juncture, ReW = 250.

Fig. 17. Flow pictures of two topologies and the skin-friction-lines. Separation saddle point (Above), attachment saddle point (Below).


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M. Y. Younis is a Ph.D. candidate at Baihang University (BUAA), Beijing, China, doing research on different as-pects of juncture flows, with main focus on topology and control of horseshoe vortex. His B.Sc, in Mechanical Engi-neering is from UET Taxila, Pakistan and M.S. in Fluid Mechanics from Bei-

hang University (BUAA), China.

investigation of different aspects of laminar horseshoe vortex system using piv

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